At 20:00 there are all the ceremonies which mark the start of Yom Hashoah. The most notable one is the one at Yad Vashem, which is the biggest in Israel, and is attended by the President and Prime Minister. At 10:00 the following morning there is a 2-minute air-raid siren throughout the country. You can hear it virtually anywhere within the state of Israel (anywhere there are air-raid sirens - which means anywhere there are people). During this siren, everybody* stops whatever it is they are doing, stands up, and pay their respect to the 6 million Jews who were murdered in the Holocaust.

Although it is a day of remembrance of the Holocaust, you will notice that it is also a day of remembrance of bravery. This means that we also remember the bravery of some Jews in the Holocaust, which stood up to the Nazis, in such places as Auschwitz and the Warsaw Ghetto.

Why the 27th of Nissan?
The date for Yom Hashoah was debated for two years before it was finally agreed upon. Many dates were suggested, including the 15th of Nissan, which is the date of the beginning of the Warsaw Ghetto Uprising. But this day landed on the date of the Passover Seder, and this was objected to. After discussions, the 27th of Nissan was chosen as it was well after passover, but still within the time of the Warsaw Ghetto uprising. Also, it falls one week before Independence day.

On April 12, 1951, the Knesset proclaimed Yom Hashoah U'Mered HaGetaot (Holocaust and Ghetto Revolt Remembrance Day) to be the 27th of Nissan. The name was later changed to what it is today. In 1959 it became a national holiday, and in 1961 the law prohibiting entertainment establishments to be open was passed.

*When I say everybody, I mean that just about everyone does. It is not a law, and some people don't. The people who don't include many minorities, such as Arabs or Christians and many Orthodox Jews, who don't accept this day. The Orthodox Jews have a problem with the Holocaust, obviously (the ages-old question - "Why would God allow the Holocaust to take place?"), and many of them don't think that we should mourn on this day, as God had His reasons. The people who DO pay their respects, stop anything they are doing. This means that when the sirens souns, all traffic stops on freeways and in the middle of intersections, and people get out of their cars and stand up for two minutes.

Sources: most of it is common knowledge in Israel. The dates are from http://history1900s.about.com/library/holocaust/aa042398.htm